that implies that
the process survives with probability 1.
Obviously this assumption could be removed, but then all results hold conditionally on
non-extinction.
Under this setting, it holds that \(\mathbb {P}(Z_n(\mathbb {Z}^d) \rightarrow \infty ) =1.\) 
Provided that \(\mathbb {E}N\log N<\infty \) , the Kesten-Stigum
theorem ([1]}, [2]}) asserts that the sequence \(W_n= Z_n(\mathbb {Z}^d)/m^n\)  converges almost surely (abbreviated as a.s.) to a finite positive random variable \( W\) .
